

We provide the semi-discrete scheme (REF ) with the discrete initial conditions \(\mathbf {u}_{h}(0)=\mathbf {u}_{h,0}\)  and \(\mathbf {B}_{h}(0)=\mathbf {B}_{h,0}\) ,
where \(\mathbf {u}_{h,0}\)  and \(\mathbf {B}_{h,0}\)  are the degrees of freedom interpolants in \(\mathbf {W}_{h}\)  and \(\mathbf {V}_{h}^{\operatorname{face}}\)  of the continuous initial conditions \(\mathbf {u}_0\)  and \(\mathbf {B}_0\)  in (REF ).
Provided that \(\mathbf {B}\)  is sufficiently regular,
from [1]}, the degrees of freedom interpolant of \(\mathbf {B}\)  in the face space \(\mathbf {V}_{h}^{\operatorname{face}}(K)\)  satisfies
\( \Vert \mathbf {B}(0)-\mathbf {B}_{h,0} \Vert _{0,K} \lesssim h_K.\) 
